//

#ifndef ABEL_RANDOM_POISSON_DISTRIBUTION_H_
#define ABEL_RANDOM_POISSON_DISTRIBUTION_H_

#include <cassert>
#include <cmath>
#include <istream>
#include <limits>
#include <ostream>
#include <type_traits>
#include "abel/random/internal/fast_uniform_bits.h"
#include "abel/random/internal/generate_real.h"
#include "abel/random/internal/iostream_state_saver.h"
#include "abel/base/math.h"

namespace abel {


// abel::poisson_distribution:
// Generates discrete variates conforming to a Poisson distribution.
//   p(n) = (mean^n / n!) exp(-mean)
//
// Depending on the parameter, the distribution selects one of the following
// algorithms:
// * The standard algorithm, attributed to Knuth, extended using a split method
// for larger values
// * The "Ratio of Uniforms as a convenient method for sampling from classical
// discrete distributions", Stadlober, 1989.
// http://www.sciencedirect.com/science/article/pii/0377042790903495
//
// NOTE: param_type.mean() is a double, which permits values larger than
// poisson_distribution<IntType>::max(), however this should be avoided and
// the distribution results are limited to the max() value.
//
// The goals of this implementation are to provide good performance while still
// beig thread-safe: This limits the implementation to not using lgamma provided
// by <math.h>.
//
template<typename IntType = int>
class poisson_distribution {
  public:
    using result_type = IntType;

    class param_type {
      public:
        using distribution_type = poisson_distribution;

        explicit param_type(double mean = 1.0);

        double mean() const { return mean_; }

        friend bool operator==(const param_type &a, const param_type &b) {
            return a.mean_ == b.mean_;
        }

        friend bool operator!=(const param_type &a, const param_type &b) {
            return !(a == b);
        }

      private:
        friend class poisson_distribution;

        double mean_;
        double emu_;  // e ^ -mean_
        double lmu_;  // ln(mean_)
        double s_;
        double log_k_;
        int split_;

        static_assert(std::is_integral<IntType>::value,
                      "Class-template abel::poisson_distribution<> must be "
                      "parameterized using an integral type.");
    };

    poisson_distribution() : poisson_distribution(1.0) {}

    explicit poisson_distribution(double mean) : param_(mean) {}

    explicit poisson_distribution(const param_type &p) : param_(p) {}

    void reset() {}

    // generating functions
    template<typename URBG>
    result_type operator()(URBG &g) {  // NOLINT(runtime/references)
        return (*this)(g, param_);
    }

    template<typename URBG>
    result_type operator()(URBG &g,  // NOLINT(runtime/references)
                           const param_type &p);

    param_type param() const { return param_; }

    void param(const param_type &p) { param_ = p; }

    result_type (min)() const { return 0; }

    result_type (max)() const { return (std::numeric_limits<result_type>::max)(); }

    double mean() const { return param_.mean(); }

    friend bool operator==(const poisson_distribution &a,
                           const poisson_distribution &b) {
        return a.param_ == b.param_;
    }

    friend bool operator!=(const poisson_distribution &a,
                           const poisson_distribution &b) {
        return a.param_ != b.param_;
    }

  private:
    param_type param_;
    random_internal::fast_uniform_bits <uint64_t> fast_u64_;
};

// -----------------------------------------------------------------------------
// Implementation details follow
// -----------------------------------------------------------------------------

template<typename IntType>
poisson_distribution<IntType>::param_type::param_type(double mean)
        : mean_(mean), split_(0) {
    assert(mean >= 0);
    assert(mean <= (std::numeric_limits<result_type>::max)());
    // As a defensive measure, avoid large values of the mean.  The rejection
    // algorithm used does not support very large values well.  It my be worth
    // changing algorithms to better deal with these cases.
    assert(mean <= 1e10);
    if (mean_ < 10) {
        // For small lambda, use the knuth method.
        split_ = 1;
        emu_ = std::exp(-mean_);
    } else if (mean_ <= 50) {
        // Use split-knuth method.
        split_ = 1 + static_cast<int>(mean_ / 10.0);
        emu_ = std::exp(-mean_ / static_cast<double>(split_));
    } else {
        // Use ratio of uniforms method.
        constexpr double k2E = 0.7357588823428846;
        constexpr double kSA = 0.4494580810294493;

        lmu_ = std::log(mean_);
        double a = mean_ + 0.5;
        s_ = kSA + std::sqrt(k2E * a);
        const double mode = std::ceil(mean_) - 1;
        log_k_ = lmu_ * mode - abel::stirling_log_factorial(mode);
    }
}

template<typename IntType>
template<typename URBG>
typename poisson_distribution<IntType>::result_type
poisson_distribution<IntType>::operator()(
        URBG &g,  // NOLINT(runtime/references)
        const param_type &p) {
    using random_internal::generate_positive_tag;
    using random_internal::generate_real_from_bits;
    using random_internal::generate_signed_tag;

    if (p.split_ != 0) {
        // Use Knuth's algorithm with range splitting to avoid floating-point
        // errors. Knuth's algorithm is: Ui is a sequence of uniform variates on
        // (0,1); return the number of variates required for product(Ui) <
        // exp(-lambda).
        //
        // The expected number of variates required for Knuth's method can be
        // computed as follows:
        // The expected value of U is 0.5, so solving for 0.5^n < exp(-lambda) gives
        // the expected number of uniform variates
        // required for a given lambda, which is:
        //  lambda = [2, 5,  9, 10, 11, 12, 13, 14, 15, 16, 17]
        //  n      = [3, 8, 13, 15, 16, 18, 19, 21, 22, 24, 25]
        //
        result_type n = 0;
        for (int split = p.split_; split > 0; --split) {
            double r = 1.0;
            do {
                r *= generate_real_from_bits<double, generate_positive_tag, true>(
                        fast_u64_(g));  // U(-1, 0)
                ++n;
            } while (r > p.emu_);
            --n;
        }
        return n;
    }

    // Use ratio of uniforms method.
    //
    // Let u ~ uniform(0, 1), v ~ uniform(-1, 1),
    //     a = lambda + 1/2,
    //     s = 1.5 - sqrt(3/e) + sqrt(2(lambda + 1/2)/e),
    //     x = s * v/u + a.
    // P(floor(x) = k | u^2 < f(floor(x))/k), where
    // f(m) = lambda^m exp(-lambda)/ m!, for 0 <= m, and f(m) = 0 otherwise,
    // and k = max(f).
    const double a = p.mean_ + 0.5;
    for (;;) {
        const double u = generate_real_from_bits<double, generate_positive_tag, false>(
                fast_u64_(g));  // U(0, 1)
        const double v = generate_real_from_bits<double, generate_signed_tag, false>(
                fast_u64_(g));  // U(-1, 1)

        const double x = std::floor(p.s_ * v / u + a);
        if (x < 0) continue;  // f(negative) = 0
        const double rhs = x * p.lmu_;
        // clang-format off
        double s = (x <= 1.0) ? 0.0
                              : (x == 2.0) ? 0.693147180559945
                                           : abel::stirling_log_factorial(x);
        // clang-format on
        const double lhs = 2.0 * std::log(u) + p.log_k_ + s;
        if (lhs < rhs) {
            return x > (max)() ? (max)()
                               : static_cast<result_type>(x);  // f(x)/k >= u^2
        }
    }
}

template<typename CharT, typename Traits, typename IntType>
std::basic_ostream<CharT, Traits> &operator<<(
        std::basic_ostream<CharT, Traits> &os,  // NOLINT(runtime/references)
        const poisson_distribution<IntType> &x) {
    auto saver = random_internal::make_ostream_state_saver(os);
    os.precision(random_internal::stream_precision_helper<double>::kPrecision);
    os << x.mean();
    return os;
}

template<typename CharT, typename Traits, typename IntType>
std::basic_istream<CharT, Traits> &operator>>(
        std::basic_istream<CharT, Traits> &is,  // NOLINT(runtime/references)
        poisson_distribution<IntType> &x) {     // NOLINT(runtime/references)
    using param_type = typename poisson_distribution<IntType>::param_type;

    auto saver = random_internal::make_istream_state_saver(is);
    double mean = random_internal::read_floating_point<double>(is);
    if (!is.fail()) {
        x.param(param_type(mean));
    }
    return is;
}


}  // namespace abel

#endif  // ABEL_RANDOM_POISSON_DISTRIBUTION_H_
